Abstract
We establish existence of periodic standing waves for a model to describe the propagation of a light pulse inside an optical fiber taking into account the Kerr effect. To this end, we apply the Lyapunov Center Theorem taking advantage that the corresponding standing wave equations can be rewritten as a Hamiltonian system. Furthermore, some of these solutions are approximated by using a Newton-type iteration, combined with a collocation-spectral strategy to discretize the system of standing wave equations. Our numerical simulations are found to be in accordance with our analytical results.
Highlights
In this paper, we consider from the theoretical and numerical point of view, periodic solutions to the system of two coupled nonlinear Schrodinger equations ∂u i ∂ξ + K ∂2u ∂x2σ1u a|u|2u g|v|2u ev2u∗ = (1) ∂v i ∂ξ ∂2v K ∂x2
We considered the existence of periodic standing wave solutions to the CNLS system (1)-(2), which is a model for several physical scenarios
It describes the propagation of a pulse along an optical fiber in the presence of nonlinearity (Kerr effect) and anomalous dispersion
Summary
We consider from the theoretical and numerical point of view, periodic solutions to the system of two coupled nonlinear Schrodinger equations ( called CNLS system). In this paper our first goal is to generalize the previous results by establishing analytically existence of periodic standing waves in the form (3) of the full system (1)-(2), considering the extra cross-mode nonlinear terms preceded by the coefficient e. Unlike previous works where topological, inverse scattering transform or quadrature techniques have been used, we apply the Lyapunov Center Theorem [20] to demonstrate analytically existence of periodic solutions to system (1)-(2) in the form (3), taking advantage that the corresponding standing wave equations can be rewritten as a Hamiltonian system.
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